The verification of probabilistic timed automata involves finding schedulers that optimise their nondeterministic choices with respect to the probability of a property. In practice, approaches based on model checking fail due to state-space explosion, while simulation-based techniques like statistical model checking are not applicable due to the nondeterminism. We present a new lightweight on-the-fly algorithm to find near-optimal schedulers for probabilistic timed automata. We make use of the classical region and zone abstractions from timed automata model checking, coupled with a recently developed smart sampling technique for statistical verification of Markov decision processes. Our algorithm provides estimates for both maximum and minimum probabilities. We compare our new approach with alternative techniques, first using tractable examples from the literature, then motivate its scalability using case studies that are intractable to numerical model checking and challenging for existing statistical techniques.